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CBSE Class 10 Answered

Show that any positive odd integer is of the form 6q+1,or 6q+3, or 6q+5, where q is some integer .
Asked by gayatrivardani2002 | 06 Jun, 2016, 12:52: AM
answered-by-expert Expert Answer

B y space E u c l i d apostrophe s space d i v i s i o n space l e m m a space a n y space p o s i t i v e space i n t e g e r space a space i s space o f space t h e space f o r m comma a equals b q plus r comma space 0 less or equal than r less than b L e t space b equals 6 T h e n space w e space h a v e space a equals 6 q plus r comma space w h e r e space 0 less or equal than r less than 6 T h u s comma space r space c a n space t a k e space v a l u e s space 0 space t o space 5 W h e n space r equals 0 comma space a equals 6 q rightwards double arrow a equals 2 cross times open parentheses 3 q close parentheses rightwards double arrow a equals 2 m comma space w h e r e space m equals 3 q T h u s comma space a space i s space a n space e v e n space i n t e g e r W h e n space r equals 1 a equals 6 q plus 1 rightwards double arrow a equals 2 cross times open parentheses 3 q close parentheses plus 1 rightwards double arrow a equals 2 m plus 1 comma space w h e r e space m equals 3 q T h u s comma space a space i s space a n space o d d space i n t e g e r W h e n space r equals 2 a equals 6 q plus 2 rightwards double arrow a equals 2 cross times open parentheses 3 q plus 1 close parentheses T h u s comma space a space i s space a n space e v e n space i n t e g e r W h e n space r equals 3 a equals 6 q plus 3 rightwards double arrow a equals 2 cross times open parentheses 3 q plus 1 close parentheses plus 1 T h u s comma space a space i s space a n space o d d space i n t e g e r W h e n space r equals 4 a equals 6 q plus 4 rightwards double arrow a equals 2 cross times open parentheses 3 q plus 2 close parentheses T h u s comma space a space i s space a n space e v e n space i n t e g e r W h e n space r equals 5 a equals 6 q plus 5 rightwards double arrow a equals 2 cross times open parentheses 3 q plus 2 close parentheses plus 1 T h u s comma space a space i s space a n space o d d space i n t e g e r A n y space i n t e g e r space c a n space b e space e i t h e r space o d d space o r space e v e n comma T h u s comma space a n y space o d d d space p o s i t i v e space i n t e g e r space i s space o f space t h e space f o r m space space 6 m plus 1 comma space 6 m plus 3 space o r space 6 m plus 5 comma space m space i s space s o m e space i n t e g e r space space space space space

Answered by | 06 Jun, 2016, 06:11: PM

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CBSE 10 - Maths
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